A variational quantum solution is to prepare a state \(| u \rangle \)  such that \(A | u \rangle \)  is proportional to a state \(| b \rangle \)  in a way that satisfies Eq. (REF ). To do that, a canonical approach [1]}, [2]}, [3]} is to first decompose the matrix \(A\)  over the Pauli basis
\(\mathcal {P}_n = \lbrace P_1\otimes \cdots \otimes P_n: \forall i,P_i \in \lbrace I,X,Y,Z\rbrace  \rbrace \)  (where \(X=|1\rangle \!\langle 0| +|0\rangle \!\langle 1|\) , \(Y = i|1\rangle \!\langle 0| -i |0\rangle \!\langle 1|\) , and \(Z=|0\rangle \!\langle 0| -|1\rangle \!\langle 1|\)  are the Pauli matrices and \(I=|0\rangle \!\langle 0| +|1\rangle \!\langle 1|\)  is the identity matrix) as
\(A=\sum _{P \in \mathcal {P}_{n}} c_{P} P,\) 
